Pricing and Hedging of Credit Derivatives via the Innovations Approach to Nonlinear Filtering R¨udiger Frey and Thorsten Schmidt June 2010 Abstract In this paper we propose a new, information-based approach for modelling the dynamic evolution of a portfolio of credit risky securities. In our setup market prices of traded credit derivatives are given by the solution o f a nonlinear filtering problem. The innovations approach to no nlinear filtering is used to so lve this problem and to der ive the dynamics of market prices. More- over, the practical application of the model is discussed: we analyse calibration, the pricing of e xotic credit derivatives and the computation of risk-minimizing hedging strategies. T he paper closes with a few numerical case studies. Keywords Credit derivatives, incomplete information, nonlinear filtering, hedging 1 Introduction Credit derivatives - derivative securities whose payoff is linked to default events in a given portfolio - are an important tool in ma naging credit risk. However, the subprime crisis and the subsequent turmoil in credit markets highlights the need for a sound methodology for the pricing and the risk management of these securities. Portfolio products pose a particular challenge in this regard: the main difficulty is to capture the dependence structure of the defaults and the dyna mic evolution o f the credit spreads in a realistic and tractable way. The authors wish to thank A. Gabih, A. Herbertsson and R. Wendler for their assistance and comments and two anonymous referees for their useful suggestions. A previous unpublished version of this paper is Frey, Gabih and Schmidt (2007). Department of Mathematics, University of Leipzig, D-04009 Leipzig, Germany. Email:ruediger.frey@math.uni-leipzig.de Department of Mathematics, Chemnitz U niversity of Technology, Reichenhainer Str. 41, D-09126 Chemnitz, Germany. Email: thorsten.schmidt@mathematik.tu-chemnitz.de 2 In this paper we propose a new, information-based approach to this prob- lem. We cons ide r a reduced-form model driven by an unobservable background factor process X. For tractability reasons X is modelled as a finite state Markov chain. We consider a market for defaultable securities related to m firms and assume that the default times are conditionally independent dou- bly stochastic random times where the default intensity of firm i is given by λ t,i = λ i (X t ). This setup is akin to the model of ?. If X was observable, the Markovian structure of the model would imply that prices of defaultable securities are functions of the past defaults and the current state of X. In our setup X is however not directly observed. Instead, the available infor- mation consists of prices of liquidly tr aded securities. Price s of such securities are given as conditional expectations with respect to a filtration F M = (F M t ) t≥0 which we call market information. We assume that F M is generated by the de- fault history of the firms under consideration and by a process Z giving obser- vations of X in additive noise. To compute the prices of the traded securities at t one therefore needs to determine the conditional distribution of X t given F M t . Since X is a finite-state Markov chain this distribution is represented by a vec tor of probabilities denoted π t . Computing the dynamics of the process π = (π t ) t≥0 is a nonlinear filter ing problem which is solved in Section 3 using martingale representation results and the innovations a pproach to nonlinear filtering. By the same token we derive the dynamics of the market price of traded credit derivatives. In Section 4 these results are then applied to the pricing and the he dging of non-traded credit derivatives. It is shown that the price of most credit derivatives common in practice - defined as conditional expectation of the associated payoff given F M t - depends on the realization of π t and on past default information. Here a major issue arises for the application of the model: we view the process Z as abstract source of informa tion which is not directly linked to economic quantities. Hence the process π is not directly accessible for typical investors. As we aim at pricing formulas a nd hedging strategies which can be eva luated in terms of publicly available information, a crucial point is to determine π t from the prices of traded sec urities (calibration), and we explain how this can be achieved by linear or quadratic programming techniques. Thereafter we derive risk-minimizing hedging strategies. Finally, in Section 5, we illustrate the applicability of the model to practical problems with a few numerical case studies . The proposed modelling approach has a number of advantages: first, ac- tual computations a re done mostly in the context of the hypothetical model where X is fully observable. Since the latter has a simple Markovian structure, computations become relatively straightforward. Second, the fact that prices of traded securities a re given by the conditional expectation given the mar- ket filtratio n F M leads to rich cre dit- spread dynamics: the proposed approach accommodates spread risk (random fluctuations of credit spreads between de- faults) and default contagion (the observation that at the default of a company the credit spreads of related companies often reac t drastically). A prime ex- ample for contagion effects is the rise in credit spreads after the default of 3 Lehman brothers in 2008. Both features are important in the derivation of robust dynamic hedging strategies and for the pricing of certain exotic credit derivatives. Third, the model has a natural factor structure with factor pr ocess π. Finally, the model calibrates reasonably well to observed market data. It is even possible to calibrate the model to single- name CDS spreads and tra nche spreads for synthetic CDOs from a heterogeneous portfolio, as is discussed in detail in Section 5.2. Reduced-form credit risk models with incomplete information have been considered previously by Sch¨onbucher (2004), Collin-Dufresne, Goldstein & Helwege (2003), Duffie, Eckner, Horel & Saita (200 9) and Frey & Runggaldier (2008). Frey & Runggaldier (2008) concentrate on the mathematical analy- sis of filtering problems in r educed-form cr edit risk models . Sch¨onbucher and Collin-Dufresne et. a l. were the fir st to point out that the successive updat- ing of the distribution of an unobservable factor in reaction to incoming de- fault observation has the potential to generate contagion effects. None of these contributions addre sses the dynamics of credit-derivative prices under incom- plete information or issues related to hedging. The innovations approach to nonlinear filtering has been used previously by Landen (2001) in the context of default-free term-structure mo de ls. Moreover, nonlinear filtering problems arise in a natural way in structural credit risk models with incomplete infor- mation about the current value of assets or liabilities such as Kusuoka (1999), Duffie & Lando (2001), Jarrow & Protter (2004), Coculescu, Geman, & Jean- blanc (2008) or Frey & Schmidt (2 009). 2 The Model Our model is constructed on some filtered probability space (Ω, F, F, Q), with F = (F t ) t≥0 satisfying the usual conditio ns; all processes considered are by assumption F-adapted. Q is the risk-neutral martingale measure used for pric- ing. For simplicity we work directly with discounted quantities so that the default-free money market account satisfies B t ≡ 1. Defaults and losses. Consider m firms. The default time of firm i is a stop- ping time de noted by τ i and the current default state of the portfolio is Y t = (Y t,1 , . . . , Y t,m ) with Y t,i = 1 {τ i ≤t} . Note that Y t ∈ {0, 1} m . We as - sume that Y 0 = 0. The percentage loss given default of firm i is denoted by the random variable ℓ i ∈ (0, 1]. We assume that ℓ 1 , . . . , ℓ m are independent random variables, independent of all other quantities introduced in the sequel. The loss state of the portfolio is given by the proces s L = (L t,1 , . . . , L t,m ) t≥0 where L t,i = ℓ i Y t,i . Marked-point-process representation. Denote by 0 = T 0 < T 1 < ··· < T m < ∞ the ordered default times and by ξ n the identity of the firm defaulting at T n . Then the sequence (T n , (ξ n , ℓ ξ n )) =: (T n , E n ), 1 ≤ n ≤ m 4 gives a r epresentation of L as marked po int process with mark space E := {1, . . . , m}×(0, 1]. Let µ L (ds, de) be the random measure associated to L with support [0, ∞) × E. Note that any random function R : Ω × [0, ∞) × E → R can be written in the form R(s, e) = R(s, (ξ, ℓ)) = m  i=1 1 {ξ=i} R i (s, ℓ) with R i (s, ℓ) := R(s, (i, ℓ)). Hence, integrals with respect to µ L (ds, de) can be written in the form t  0  E R(s, e) µ L (ds, de) =  T n ≤t R ξ n (T n , ℓ ξ n ) =  τ i ≤t R i (τ i , ℓ i ). (2.1) 2.1 The underlying Markov model The default intensities of the firms under consideration are driven by the so-called factor or state process X. The process X is modelled as a finite- state Markov chain; in the sequel its state space S X is identified with the set {1, . . . , K}. The following assumption sta tes that the default times are condi- tionally independent, doubly -stochastic random times with default intensity λ t,i := λ i (X t ). Set F X ∞ = σ(X s : s ≥ 0). A1 There are functions λ i : S X → (0, ∞), i = 1, . . . , m, such that for all t 1 , . . . , t m ≥ 0 Q  τ 1 > t 1 , . . . , τ m > t m | F X ∞  = m  i=1 exp  − t i  0 λ i (X s )ds  . It is well-known that under A1 there are no joint defaults, i.e. τ i = τ j , for i = j almo st surely. Moreover, for all 1 ≤ i ≤ m Y t,i − t∧τ i  0 λ i (X s )ds (2.2) is an F-martingale; see for instance Cha pter 9 in McNeil, Frey & Embrechts (2005). Furthermore, the process (X, L) is jointly Markov. Denote by F ℓ i the distribution function of ℓ i . A default of firm i occurs with intensity (1 − Y t,i )λ i (X t ), and the loss given default of firm i has the distribution F ℓ i . Hence the F-compensator ν L of the random measure µ L is given by ν L (dt, de) = ν L (dt, dξ, dℓ) = m  i=1 δ {i} (dξ) F ℓ i (dℓ) (1 − Y t,i )λ i (X t )dt , (2.3) 5 where δ {i} stands for the Dirac-measure in i. To illustrate this further, we show how the default intensity of company j can be r e covered from (2.3): note that Y t,j = 1 {τ j ≤t} =  T n ≤t 1 {ξ n =j} = t  0  E R j (s, e)µ L (ds, de) with R j (s, e) = R j (s, (ξ, ℓ)) := 1 {ξ=j} . Using (2.1), the compensator of Y j is given by t  0  E R j (s, e)ν L (ds, de) = t  0  E 1 {ξ=j} m  i=1 δ {i} (dξ) F ℓ i (dℓ) (1 − Y s,i )λ i (X s )ds = t  0 (1 − Y s,j )λ j (X s )ds. Example 2.1 In the numerical pa rt we will consider a one-factor model where X represents the global state of the economy. For this we model the default intensities under full information as increasing functions λ i : {1, . . . , K} → (0, ∞). Hence, 1 represents the best state (lowest default intensity) and K corres ponds to the worst state; moreover, the default intensities are comono- tonic. In the special case of a homogeneous model the default intensities of all firms are identical, λ i (·) ≡ λ(·). Furthermore , denote by (q(i, k)) 1≤i,k≤K the generator matrix of X so that q(i, k), i = k, gives the intensity o f a trans ition from state i to state k. We will consider two possible choices for this matrix. First, let the factor process be constant, X t ≡ X for all t. In that case q(i, k) ≡ 0, and filtering r educes to Bayesian analysis. A model of this type is known as frailty model, see also Sch¨onbucher (2004). Second, we consider the case where X has next neighbour dynamics, that is, the chain jumps from X t only to the neig hbouring points X t + − 1 (with the obvious modifications for X t = 0 and X t = K). 2.2 Ma rket information In our setting the factor process X is not directly observable. We assume that prices of traded credit derivatives are determined as conditional expectation with respect to some filtration F M which we call market information. The following assumption states that F M is generated by the loss history F L and observations of functions of X in additive Gaussia n noise. A2 F M = F L ∨ F Z , where the l-dimensional process Z is given by Z t = t  0 a(X s ) ds + B t . (2.4) Here, B is an l-dimensional standard F-Brownian motion independent of X and L, and a(·) is a function from S X to R l . 6 In the case of a homogeneous model one could take l = 1 and assume that a(·) = c ln λ(·). Here the constant c ≥ 0 models the info rmation-content of Y : for c = 0, Y carries no information, whe reas for c large the state X t can be observed with high precisio n. 3 Dynamics of traded credit derivatives and filtering In this section we study in detail traded credit derivatives. First, we give a general description of this type of derivatives and discuss the relatio n between pricing and filtering. In Section 3.2 we then study the dynamics of market prices, using the innovations approach to nonlinear filtering. 3.1 Traded securities We consider a market of N liquidly traded credit derivatives, with - fo r no- tational simplicity - common maturity T . Most credit derivatives have inter- mediate cash flows such as payments at default dates and it is c onvenient to describe the payoff of the nth derivative by the cumulative dividend stream D n . We assume that D n takes the form D t,n = t  0 d 1,n (s, L s )db(s) + t  0  E d 2,n (s, L s− , e)µ L (ds, de) (3.1) with bounded functions d 1 , d 2 and an increasing deterministic function b : [0, T ] → R. Dividend streams o f the fo rm (3.1) c an be used to model many important credit derivatives, as the following examples show. Zero-bond. A defaultable bond on firm i without coupon payments and with zero recovery pays 1 at T if τ i > T and zero otherwise. Hence, we have b(s) = 1 {s≥T } , d 1 (t, L t ) = 1 {L t,i =0} and d 2 = 0. For CDS and CDO the function b encodes the pre-scheduled payments: fo r payment dates t 1 < ··· < t ˜n < T we set b(s) = |{i: t i ≤ s}|. Credit default swap (CDS). A protection seller position in a CDS on firm i offers regular payments of size S at t 1 , . . . , t ˜n until default. In exchange for this , the holder pays the loss ℓ i at τ i , provided τ i < T (accrued pre- mium payments are ignored for simplicity). This can be mo de lled by taking d 1 (t, L t ) = S1 {L t,i =0} and d 2 (t, L t− , (ξ, ℓ)) = −1 {t≤T } 1 {ξ=i} ℓ; note that t  0  E d 2,n (s, L s− , e)µ L (ds, de) = −ℓ i 1 {L t,i >0} = −L t,i . Collateralized debt obligation (CDO). A single tranche CDO on the un- derlying portfolio is specified by an lowe r and upper detachment point 1 0 ≤ 1 In practice, lower and upper detachment points are stated in percentage points, say 0 ≤ l < u ≤ 1. Then x 1 = l · m and x 2 = u · m. 7 x 1 < x 2 ≤ m and a fixed spr ead S. Denote the cumu lative port folio loss by ¯ L t =  m i=1 L t,i , and define the function H(x) := (x 2 − x) + − (x 1 − x) + . An investor in a CDO tranche receives at payment date t i a spread payment proportional to the remaining notional H( ¯ L t i ) of the tranche. Hence, his in- come stream is given by  t 0 SH( ¯ L s )db(s), s o that d 1 (t, L t ) = SH( ¯ L t ). In return the investor pays at the successive default times T n with T n ≤ T the amount −∆H( ¯ L T n ) = −  H( ¯ L T n ) − H( ¯ L T n − )  (the part of the portfolio loss falling in the tranche). This can be modelled by setting d 2 (t, L t− , (ξ, ℓ)) = 1 {t≤T } H  ℓ + ¯ L t−  − H  ¯ L t−  . Other credit derivatives such as CDS indices or typical basket swaps can be modelled in a similar way. Pricing of traded credit derivatives. Recall that we work with discounted quan- tities, that Q represents the underlying pricing measure, and the information available to market par ticipants is the market information F M . As a conse- quence we assume that the current market value of the traded credit deriva- tives is given by p t,n := E  D T,n − D t,n |F M t  , 1 ≤ n ≤ N. (3.2) The gains process g n of the n-th credit derivative sums the current market value and the dividend payments received so far and is thus given by g t,n := p t,n + D t,n = E  D T,n | F M t  ; (3.3) in particular, g n is a martingale. Next, we show that the computation of market values leads to a nonlinear filtering problem. We call E  D T,n − D t,n |F t  the hypothetical value of D n . While this quantity will be an important tool in our analysis it does not corres pond to market pr ices as in contrast to p n it is not F M -adapted. Observe that by (3.1) D T,n − D t,n is a function of the future path (L s ) s∈(t,T ] . Hence, the F-Markov property of the pair (X, L) implies that E  D T,n − D t,n |F t  = p n (t, X t , L t ) (3.4) for functions p n : [0, T ] × S X × [0, 1] m → R, n = 1, . . . , N; see for instance Proposition 2.5.15 in Karatzas & Shreve (1988) for a general version of the Markov property that covers (3.4). By iterated conditional expectations we obtain p t,n = E  E  D T,n − D t,n |F t  |F M t  = E  p n (t, X t , L t )|F M t  . (3.5) In order to compute the market values p t,n we therefore need to determine the conditional distribution of X t given F M t . This a nonlinea r filtering problem which we solve in Section 3.3 below. 8 Remark 3.1 (Computation of the full-information value) For bo nds and CDSs the evaluation of p n can be done via the Feynman-Kac formula and related Markov chain techniques; for instance see Elliott & Mamon (2003). In the case of CDOs, the evaluation of p n via Laplace transforms is discussed in ?. Alternatively, a two stage method that employs the conditional independence of defaults given F X ∞ can be used. For this, one first generates a trajectory of X. Given this trajectory, the loss distributio n can then be evaluated using one of the known methods for computing the distribution of the sum of inde- pendent (but not identically distributed) Bernoulli var iables. Finally, the loss distribution is estimated by ave raging over the sampled trajectories of X. An extensive numerical case study comparing the different approaches is given in Wendler (2010). 3.2 Asse t price dynamics under the market filtration In the sequel we use the innovations approach to nonlinear filtering in o rder to derive a repre sentation o f the martingales g n as a stochastic integral with respect to certain F M -adapted martingales. Fo r a generic process U we denote by  U t := E(U t |F M t ) the optional projection of U w.r.t. the market filtration F M in the rest of the paper. Moreover, for a generic function f : S X → R we use the abbreviation  f for the optiona l projection of the process (f(X s )) s≥0 with respect to F M . We begin by introducing the martingales neede d for the repr esentation result. Firs t, define for i = 1, . . . , l m Z t,i := Z t,i − t  0 ( a i ) s ds . (3.6) It is well-known that m Z is an F M -Brownian motion and thus the martingale part in the F M -semimartingale decomposition of Z. Second, denote by ν L (dt, de) := m  i=1 δ {i} (dξ) F ℓ i (dℓ) (1 − Y t,i )(  λ i ) t dt (3.7) the comp ensator of µ L w.r.t. F M and define the compensated random measure m L (dt, de) := µ L (dt, de) − ν L (dt, de) . (3.8) Corollary VIII.C4 in Br´emaud (1981) yields that for every F M -predictable random function f such that E   E  T 0 |f(s, e)|ν L (ds, de)  < ∞ the integral  E  t 0 f(s, e) m L (ds, de) is a martingale with respect to F M . The following ma rtingale representation r esult is a key tool in our analysis; its proof is relegated to the appendix. 9 Lemma 3.2 For every F M -martingale (U t ) 0≤t≤T there exists a F M -predictable function γ : Ω × [0, T ] × E → R and an R l -valued F M -adapted process α satisfying  T 0 ||α s || 2 ds < ∞ Q-a.s. and  T 0  E |γ(s, e)|ν L (ds, de) < ∞ Q-a.s. such that U has the representation U t = U 0 + t  0  E γ(s, e) m L (ds, de) + t  0 α ⊤ s dm Z s , 0 ≤ t ≤ T. (3.9) The nex t theorem is the basis for the mathematical analysis of the model under the market filtration. Theorem 3.3 Consider a real-valued F-semimartingale J t = J 0 + t  0 A s ds + M J t , t ≤ T such that [M J , B] = 0. Assume that (i) E(|J 0 |) < ∞, E(  T 0 |A s |ds) < ∞ and E(  T 0 |J s |λ i (X s )ds) < ∞, 1 ≤ i ≤ m. (ii) E([M J ] T ) < ∞. (iii) For all 1 ≤ i ≤ m there is some F M -predictable R i : Ω ×[0, T ] ×(0 , 1] → R such that [J, Y i ] t = t  0  E 1 {ξ=i} R i (s, ℓ) µ L (ds, dξ, dℓ). (3.10) Moreover, E(  T 0  1 0 |R i (s, ℓ)|F ℓ i (dℓ)(1 − Y s,i )λ i (X s )ds) < ∞. (iv)  t 0 J s dB s,j and  t 0 Z s,j dM J s , 1 ≤ j ≤ l are true F-martingales. Then the optional projection  J has the representation  J t =  J 0 + t  0  A s ds + t  0  E γ(s, e) m L (ds, de) + t  0 α ⊤ s dm Z s , t ≤ T ; (3.11) here, γ(s, e) = γ(s, (ξ, ℓ)) =  m i=1 1 {ξ=i} γ i (s, ℓ), and α, γ i are given by α s = (  Ja) s −  J s (  a) s , (3.12) γ i (s, ℓ) = 1 (  λ i ) s−  (  Jλ i ) s− −  J s− (  λ i ) s− + (  R i (·, ℓ)λ i ) s−  . (3.13) Proof The proof uses the following two well-known facts. 1. For every true F-martingale N , the projection  N is an F M -martingale. 2. For any progressively measurable process φ w ith E  T 0 |φ s |ds  < ∞ the process   t 0 φ s ds −  t 0  φ s ds, t ≤ T , is an F M -martingale. 10 The first fact is simply a consequenc e of iterated expectations, while the second follows fr om the Fubini theorem, see for instance Davis & Marcus (1981). As M J is a true martingale by (ii), Fact 1 and 2 immediately yield that  J t −  J 0 −  t 0  A s ds is an F M -martingale. Lemma 3.2 thus gives the e xistence of the r epresentation (3.11). It remains to identify γ and α. The idea is to use the elementary identity  Jφ =  J φ for any F M -adapted φ. Each side of this equation gives rise to a different semimartingale decomposition of  Jφ ; comparing those for suitably chosen φ one obtains γ and α. In order to identify γ, fix i and let φ i t = t  0  E ϕ(s, ℓ)1 {ξ=i} µ L (ds, dξ, dℓ) for a bounded and F M -predictable ϕ. Note that φ i is F M -adapted. We first determine the F-semimartingale decomposition of J φ i . Itˆo’s formula gives d(J t φ i t ) = φ i t− dJ t + J t− dφ i t + d[J, φ i ] t . (3.14) With (3.10), [J, φ i ] t =  s≤t ∆J s ∆φ i s = t  0  E R i (s, ℓ)ϕ(s, ℓ)1 {ξ=i} µ L (ds, dξ, dℓ). Hence, using (2.3), the predictable compensator of [J, φ i ] is J, φ i  t = t  0 1  0 R i (s, ℓ)ϕ(s, ℓ)F ℓ i (dℓ)(1 − Y s,i )λ i (X s )ds. (3.15) Moreover, [J, φ i ] −J, φ i  is a true martingale by (iii), as ϕ is bounded. Using (3.14) and (3.15) the finite variation part in the F-semima rtingale decomposi- tion of Jφ i =: ˜ A + ˜ M computes to ˜ A t = t  0  φ i s A s + J s (1 − Y s,i )λ i (X s ) 1  0 ϕ(s, ℓ)F ℓ i (dℓ) + 1  0 R i (s, ℓ)ϕ(s, ℓ)(1 − Y s,i )λ i (X s )F ℓ i (dℓ)  ds. [...]... , the current value of the process π On the other hand, the model parameters (the generator matrix of X and parameters of the functions a(·) and λi (·), i = 1, , m) need to be estimated The latter task depends on the specific parametrization of the model and on the available data We discuss parameter estimation for the frailty model in Section 5 Here we concentrate on the determination of π t The. .. only on the current market state (Lt , πt ) and on the function p(·) that gives the hypothetical value of the option in the underlying Markov model Hence the precise 2 The evaluation of p(·) can be done with similar methods as in Remark 3.1 16 form of the function a(·) from A2 and thus of the dynamics of π is irrelevant for the pricing of these claims; the dynamics of π do however matter in the computation... hedge ratio θ for hedging a CDO tranche with the underlying CDS index in the homogeneous version of the frailty model The numbers were computed using the probability vector π ∗ obtained via calibration to the iTraxx data from 2006 and 2009 6 For this we used (3.19) to determine the diffusion part in the dynamics of the default leg Vtdef and the premium leg Vtprem The diffusion part of the spread St =... this section we discuss the pricing, the calibration, and the hedging of credit derivatives Consider a non-traded credit derivative In accordance with (3.2), we define the price at time t of the credit derivative as conditional expectation M of the associated payoff given Ft For the credit derivatives common in practice this conditional expectation is given by a function of the current market state... Upon exercise the owner of the option holds a protection-buyer position on the underlying index with a pre-specified spread S (the exercise spread of the option); moreover, he obtains the cumulative portfolio loss up to time U Denote by V def (t, Xt , Lt ) and V prem (t, Xt , Lt ) the full-information value of the default and the premium payment leg of the CDS index In our setup the value of the option... used to insure integrability conditions in Theorem 3.3 and in Theorem 3.4 so that these results are easily extended to a more general setting The filtering results in Section 3.3 below on the other hand do exploit the specific structure of X 3.3 Filtering and factor representation of market prices Since X is a finite state Markov chain, the conditional distribution of Xt given M 1 K k M Ft is given by the. .. note that the arising local martingales in the semimartingale decomposition of JZi are true martingales by (iv) ⊓ ⊔ The following theorem describes the dynamics of the gains processes of the traded credit derivatives and gives their instantaneous quadratic covariation Theorem 3.4 Under A1 and A2 the gains processes g1 , , gN of the traded securities have the martingale representation m t t g 1{ξ=i}... where ei stands for the i-th unit vector in Rm The FM -compensator of the random measure µS associated with the jumps of S under Q∗ is m ν S (dt, d˜) := e 1{˜=(0,ei ,ℓei )} Fℓi (dℓ) (1 − Yt−,i )dt e i=1 Theorem III.2.34 Jacod & Shiryaev (2003) now shows that the martingale problem associated with the characteristics of S has a unique solution; Theorem III.4.29 of the same source then gives that the Q∗... Derivatives (to appear)’ Wiley, New York Frey, R., Schmidt, T & Gabih, A (2007), Pricing and hedging of credit derivatives via nonlinear filtering’, preprint, Universit¨t Leipzig a Graziano, G & Rogers, C (2009), ‘A dynamic approach to the modelling of correlation credit derivatives using Markov chains’, 12, 45–62 Hull, J & White, A (2006), ‘Valuing credit derivatives using an implied copula approach , The. .. applied to the outcome of Step 4 (see (4.6)) In the homogeneous case (i.e d1 = · · · = dm ) the parameter d1 can be kept fixed during the calibration Calibration results We present results from two types of numerical experiments5 First, we calibrated the homogeneous version of the model to tranche and index spreads from the iTraxx Europe in the years 2006 (before the credit crisis) and 2009 The calibration . the same token we derive the dynamics of the market price of traded credit derivatives. In Section 4 these results are then applied to the pricing and the he dging of non-traded credit derivatives. . current time t one needs to determine π t , the current value of the process π. On the other hand, the model parameters (the generator matrix of X and para meters of the functions a(·) and λ i (·), i =. L t ) the full-information value of the default and the pr emium payment leg of the CDS index. In our setup the value of the option at maturity U is then given by the following function of the